Inmathematics, more specificallyalgebra,abstract algebra ormodern algebra is the study ofalgebraic structures, which aresets with specificoperations acting on their elements.[1] Algebraic structures includegroups,rings,fields,modules,vector spaces,lattices, andalgebras over a field. The termabstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically fromelementary algebra, the use ofvariables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except inpedagogy.
Algebraic structures, with their associatedhomomorphisms, formmathematical categories.Category theory gives a unified framework to study properties and constructions that are similar for various structures.
Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called thevariety of groups.
Before the nineteenth century,algebra was defined as the study ofpolynomials.[2] Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed. Concrete problems and examples came from number theory, geometry, analysis, and the solutions ofalgebraic equations. Most theories that are now recognized as parts of abstract algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. This unification occurred in the early decades of the 20th century and resulted in the formalaxiomatic definitions of variousalgebraic structures such as groups, rings, and fields.[3] This historical development is almost the opposite of the treatment found in popular textbooks, such as van der Waerden'sModerne Algebra,[4] which start each chapter with a formal definition of a structure and then follow it with concrete examples.[5]
The study of polynomial equations oralgebraic equations has a long history. Circa 1700 BC, the Babylonians were able to solve quadratic equations specified as word problems. This word problem stage is classified asrhetorical algebra and was the dominant approach up to the 16th century.Al-Khwarizmi originated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. Fully symbolic algebra did not appear untilFrançois Viète's 1591New Algebra, and even this had some spelled out words that were given symbols in Descartes's 1637La Géométrie.[6] The formal study of solving symbolic equations ledLeonhard Euler to accept what were then considered "nonsense" roots such asnegative numbers andimaginary numbers, in the late 18th century.[7] However, European mathematicians, for the most part, resisted these concepts until the middle of the 19th century.[8]
George Peacock's 1830Treatise of Algebra was the first attempt to place algebra on a strictly symbolic basis. He distinguished a newsymbolical algebra, distinct from the oldarithmetical algebra. Whereas in arithmetical algebra is restricted to, in symbolical algebra all rules of operations hold with no restrictions. Using this Peacock could show laws such as, by letting in. Peacock used what he termed theprinciple of the permanence of equivalent forms to justify his argument, but his reasoning suffered from theproblem of induction.[9] For example, holds for the nonnegativereal numbers, but not for generalcomplex numbers.
The abstract concept of group emerged slowly over the middle of the nineteenth century. Galois in 1832 was the first to use the term "group",[11] signifying a collection of permutations closed under composition.[12]Arthur Cayley's 1854 paperOn the theory of groups defined a group as a set with an associative composition operation and the identity 1, today called amonoid.[13] In 1870 Kronecker defined an abstract binary operation that was closed, commutative, associative, and had the leftcancellation property,[14] similar to the modern laws for a finiteabelian group.[15] Weber's 1882 definition of a group was a closed binary operation that was associative and had left and right cancellation.[16]Walther von Dyck in 1882 was the first to require inverse elements as part of the definition of a group.[17]
Once this abstract group concept emerged, results were reformulated in this abstract setting. For example,Sylow's theorem was reproven by Frobenius in 1887 directly from the laws of a finite group, although Frobenius remarked that the theorem followed from Cauchy's theorem on permutation groups and the fact that every finite group is a subgroup of a permutation group.[18][19]Otto Hölder was particularly prolific in this area, defining quotient groups in 1889, group automorphisms in 1893, as well as simple groups. He also completed theJordan–Hölder theorem. Dedekind and Miller independently characterizedHamiltonian groups and introduced the notion of thecommutator of two elements. Burnside, Frobenius, and Molien created therepresentation theory of finite groups at the end of the nineteenth century.[18] J. A. de Séguier's 1905 monographElements of the Theory of Abstract Groups presented many of these results in an abstract, general form, relegating "concrete" groups to an appendix, although it was limited to finite groups. The first monograph on both finite and infinite abstract groups was O. K. Schmidt's 1916Abstract Theory of Groups.[20]
Once there were sufficient examples, it remained to classify them. In an 1870 monograph,Benjamin Peirce classified the more than 150 hypercomplex number systems of dimension below 6, and gave an explicit definition of anassociative algebra. He defined nilpotent and idempotent elements and proved that any algebra contains one or the other. He also defined thePeirce decomposition. Frobenius in 1878 andCharles Sanders Peirce in 1881 independently proved that the only finite-dimensional division algebras over were the real numbers, the complex numbers, and the quaternions. In the 1880s Killing and Cartan showed that semisimpleLie algebras could be decomposed into simple ones, and classified all simple Lie algebras. Inspired by this, in the 1890s Cartan, Frobenius, and Molien proved (independently) that a finite-dimensional associative algebra over or uniquely decomposes into thedirect sums of a nilpotent algebra and a semisimple algebra that is the product of some number ofsimple algebras, square matrices over division algebras. Cartan was the first to define concepts such as direct sum and simple algebra, and these concepts proved quite influential. In 1907 Wedderburn extended Cartan's results to an arbitrary field, in what are now called theWedderburn principal theorem andArtin–Wedderburn theorem.[25]
In the 1850s, Riemann introduced the fundamental concept of aRiemann surface. Riemann's methods relied on an assumption he calledDirichlet's principle,[30] which in 1870 was questioned by Weierstrass. Much later, in 1900, Hilbert justified Riemann's approach by developing thedirect method in the calculus of variations.[31] In the 1860s and 1870s, Clebsch, Gordan, Brill, and especiallyM. Noether studiedalgebraic functions and curves. In particular, Noether studied what conditions were required for a polynomial to be an element of the ideal generated by two algebraic curves in the polynomial ring, although Noether did not use this modern language. In 1882 Dedekind and Weber, in analogy with Dedekind's earlier work on algebraic number theory, created a theory ofalgebraic function fields which allowed the first rigorous definition of a Riemann surface and a rigorous proof of theRiemann–Roch theorem. Kronecker in the 1880s, Hilbert in 1890, Lasker in 1905, andMacaulay in 1913 further investigated the ideals of polynomial rings implicit inE. Noether's work. Lasker proved a special case of theLasker-Noether theorem, namely that every ideal in a polynomial ring is a finite intersection ofprimary ideals. Macaulay proved the uniqueness of this decomposition.[32] Overall, this work led to the development ofalgebraic geometry.[26]
In 1801 Gauss introducedbinary quadratic forms over the integers and defined theirequivalence. He further defined thediscriminant of these forms, which is aninvariant of a binary form. Between the 1860s and 1890sinvariant theory developed and became a major field of algebra. Cayley, Sylvester, Gordan and others found theJacobian and theHessian for binary quartic forms and cubic forms.[33] In 1868 Gordan proved that thegraded algebra of invariants of a binary form over the complex numbers was finitely generated, i.e., has a basis.[34] Hilbert wrote a thesis on invariants in 1885 and in 1890 showed that any form of any degree or number of variables has a basis. He extended this further in 1890 toHilbert's basis theorem.[35]
Once these theories had been developed, it was still several decades until an abstract ring concept emerged. The first axiomatic definition was given byAbraham Fraenkel in 1914.[35] His definition was mainly the standard axioms: a set with two operations addition, which forms a group (not necessarily commutative), and multiplication, which is associative, distributes over addition, and has an identity element.[36] In addition, he had two axioms on "regular elements" inspired by work on thep-adic numbers, which excluded now-common rings such as the ring of integers. These allowed Fraenkel to prove that addition was commutative.[37] Fraenkel's work aimed to transfer Steinitz's 1910 definition of fields over to rings, but it was not connected with the existing work on concrete systems. Masazo Sono's 1917 definition was the first equivalent to the present one.[38]
In 1920,Emmy Noether, in collaboration with W. Schmeidler, published a paper about thetheory of ideals in which they definedleft and right ideals in aring. The following year she published a landmark paper calledIdealtheorie in Ringbereichen (Ideal theory in rings'), analyzingascending chain conditions with regard to (mathematical) ideals. The publication gave rise to the term "Noetherian ring", and several other mathematical objects being calledNoetherian.[39][40] Noted algebraistIrving Kaplansky called this work "revolutionary";[39] results which seemed inextricably connected to properties of polynomial rings were shown to follow from a single axiom.[41] Artin, inspired by Noether's work, came up with thedescending chain condition. These definitions marked the birth of abstract ring theory.[42]
In 1801 Gauss introduced theintegers mod p, where p is a prime number. Galois extended this in 1830 tofinite fields with elements.[43] In 1871Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations,[44] theGerman wordKörper, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore in 1893.[45] In 1881Leopold Kronecker defined what he called adomain of rationality, which is a field ofrational fractions in modern terms.[46] The first clear definition of an abstract field was due toHeinrich Martin Weber in 1893. It was missing the associative law for multiplication, but covered finite fields and the fields of algebraic number theory and algebraic geometry.[47] In 1910 Steinitz synthesized the knowledge of abstract field theory accumulated so far. He axiomatically defined fields with the modern definition, classified them by theircharacteristic, and proved many theorems commonly seen today.[48]
The end of the 19th and the beginning of the 20th century saw a shift in the methodology of mathematics. Abstract algebra emerged around the start of the 20th century, under the namemodern algebra. Its study was part of the drive for moreintellectual rigor in mathematics. Initially, the assumptions in classicalalgebra, on which the whole of mathematics (and major parts of thenatural sciences) depend, took the form ofaxiomatic systems. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certainalgebraic structures began to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of anabstract group. Questions of structure and classification of various mathematical objects came to the forefront.[50]
These processes were occurring throughout all of mathematics but became especially pronounced in algebra. Formal definitions through primitive operations and axioms were proposed for many basic algebraic structures, such asgroups,rings, andfields. Hence such things asgroup theory andring theory took their places inpure mathematics. The algebraic investigations of general fields byErnst Steinitz and of commutative and then general rings byDavid Hilbert,Emil Artin andEmmy Noether, building on the work ofErnst Kummer,Leopold Kronecker andRichard Dedekind, who had considered ideals in commutative rings, and ofGeorg Frobenius andIssai Schur, concerningrepresentation theory of groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of the 20th century were systematically exposed inBartel van der Waerden'sModerne Algebra, the two-volumemonograph published in 1930–1931 that reoriented the idea of algebra fromthe theory of equations tothetheory of algebraic structures.[51]
By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied aresets, to which the theorems ofset theory apply. Those sets that have a certain binary operation defined on them formmagmas, to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to formsemigroups); identity, and inverses (to formgroups); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems ofgroup theory may be used when studyingrings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. In general there is a balance between the amount of generality and the richness of the theory: more general structures have usually fewernontrivial theorems and fewer applications.[citation needed]
Algebraic structures betweenmagmas andgroups. For example, monoids aresemigroups with identity.
Examples of algebraic structures with a singlebinary operation are:
A group is a set together with a "group product", a binary operation. The group satisfies the following defining axioms (cf.Group (mathematics) § Definition):
Identity: there exists an element such that, for each element in, it holds that.
Inverse: for each element of, there exists an element so that.
Associativity: for each triplet of elements in, it holds that.
Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance,algebraic topology uses algebraic objects to study topologies. ThePoincaré conjecture, proved in 2003, asserts that thefundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not.Algebraic number theory studies various numberrings that generalize the set of integers. Using tools of algebraic number theory,Andrew Wiles provedFermat's Last Theorem.[citation needed]
In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. Ingauge theory, the requirement oflocal symmetry can be used to deduce the equations describing a system. The groups that describe those symmetries areLie groups, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number offorce carriers in a theory is equal to the dimension of the Lie algebra, and thesebosons interact with the force they mediate if the Lie algebra is nonabelian.[52]
^Euler, Leonhard (1748).Introductio in Analysin Infinitorum [Introduction to the Analysis of the Infinite] (in Latin). Vol. 1. Lucerne, Switzerland: Marc Michel Bosquet & Co. p. 104.
^Martinez, Alberto (2014).Negative Math. Princeton University Press. pp. 80–109.
^Monna 1975, pp. 55–56, citingHilbert, David (1905), "Über das Dirichletsche Prinzip",Journal für die reine und angewandte Mathematik (in German), vol. 129, pp. 63–67
^Corry, Leo."Structural algebra". Britannica.Archived from the original on 16 December 2024. Retrieved10 February 2025.Then in 1930 a textbook was published that presented a totally new image of the discipline. This was Moderne Algebra, by the Dutch mathematician Bartel van der Waerden, who since 1924 had attended lectures in Germany by Emmy Noether at Göttingen and by Emil Artin at Hamburg. Van der Waerden's new image of the discipline inverted the conceptual hierarchy of classical algebra. Groups, fields, rings, and other related concepts became the main focus, based on the implicit realization that all of these concepts were, in fact, instances of a more general, underlying idea: the idea of an algebraic structure.
Kimberling, Clark (1981). "Emmy Noether and Her Influence". In Brewer, James W; Smith, Martha K (eds.).Emmy Noether: A Tribute to Her Life and Work.Marcel Dekker. pp. 3–61.
Monna, A. F. (1975),Dirichlet's principle: A mathematical comedy of errors and its influence on the development of analysis, Oosthoek, Scheltema & Holkema,ISBN978-9031301751
